I don't expect that there's a clear-cut answer to any of my quite detailed and tricky questions, but I'll give it a try.
1) This female-type House Sparrow looks like it has brown neck sides, at least somewhat brown.
... in wilderness _____________________ House Sparrow ... female Village Mandoti Jhajjar Haryana India 8 August 2023
www.birdforum.net
This female-type bird was identified as a vagrant Sind Sparrow. It has some brown but not as contrasting as the female House Sparrow above.
Macaulay Library ML294121471; © Ranjeet Singh; Sikar, Rajasthan, India
macaulaylibrary.org
Is it possible that both of these IDs are correct**?
2) Both Opus and eBird say the male has a black bill. This is not exactly the case in eBird's own featured photos, in which only the tip is black:
Macaulay Library ML132809941; © Saurabh Sawant; Ferozepur, Punjab, India
macaulaylibrary.org
Macaulay Library ML118787031; © Kavi Nanda; Gurugram, Haryana, India
macaulaylibrary.org
The photo below was taken some 100 km to the northwest of Jaipur (out of the described range*) and shows a male with an all pale bill. Is it mis-identified?
Macaulay Library ML297440401; © Ranjeet Singh; Sikar, Rajasthan, India
macaulaylibrary.org
On the other hand, the sparrow in the photo below also has a pale bill, so maybe it's the correct ID, only the feature is nonexistent. And, anyway, the male from the eBird record above has a certain non-House Sparrow feel to it.
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ogaclicks.com
I think I remember reading that House Sparrow changes bill colour in breeding plumage, so that could be a plausible explanation for Sind Sparrow as well. The source above, which is quite reliable*** (I think) also apparently endorses this notion.
*EDIT: It's still 100 km, and it's just a vagrant, so there is still no realistic chance of it occuring in Jaipur proper.
***EDIT 3: But why does the sparrow in the picture at the top of the blog post lack the grey cheek--just a House Sparrow after all?
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ogaclicks.com
**EDIT 2: Probably the first sparrow has too little brown on the neck, but--if it does--then the other one has no more of it, which would make the second bird a House Sparrow as well. Unless the cheek of the second bird counts as grey.
One more picture of an immature as a bonus.
www.birdforum.net
I do wonder if they don't hybridise with each other somewhere, given that much more dissimilar sparrow species do.